# Singular Solutions of Elliptic Equations with Iterated Exponentials

@article{Ghergu2019SingularSO,
title={Singular Solutions of Elliptic Equations with Iterated Exponentials},
author={Marius Ghergu and Olivier Goubet},
journal={The Journal of Geometric Analysis},
year={2019},
volume={30},
pages={1755-1773}
}
• Published 12 June 2019
• Mathematics
• The Journal of Geometric Analysis
We construct positive singular solutions for the problem $$-\Delta u=\lambda \exp (e^u)$$ - Δ u = λ exp ( e u ) in $$B_1\subset {\mathbb {R}}^n$$ B 1 ⊂ R n ( $$n\ge 3$$ n ≥ 3 ), $$u=0$$ u = 0 on $$\partial B_1$$ ∂ B 1 , having a prescribed behaviour around the origin. Our study extends the one in Miyamoto (J Differ Equ 264:2684–2707, 2018) for such nonlinearities. Our approach is then carried out to elliptic equations featuring iterated exponentials.
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We investigate the structure of radial solutions corresponding to the equation $\Delta u=\frac{1}{f(u)}\ \ \textrm{in}\ B_{r_0}\subset\mathbb{R}^N,\ N\ge 3,\ r_0>0,$ where $f\in C[0,\infty)\cap Radial single point rupture solutions for a general MEMS model • Mathematics Calculus of Variations and Partial Differential Equations • 2022 We study the initial value problem $$\begin{cases} r^{-(\gamma-1)}\left(r^{\alpha}|u'|^{\beta-1}u'\right)'=\frac{1}{f(u)} & \textrm{for}\ 0 0 & \textrm{for}\ 0 \alpha>\beta\geq 1 and f\in C[0,\bar ## References SHOWING 1-10 OF 25 REFERENCES Perturbing singular solutions of the Gelfand problem • Mathematics • 2007 he equation -\Delta u = \lambda e^u posed in the unit ball B \subseteq \R^N, with homogeneous Dirichlet condition u|_{\partial B} = 0, has the singular solution U=\log\frac1{|x|^2} when Partial regularity for a Liouville system • Mathematics • 2013 Let \Omega\subset\mathbb{R}^n be a bounded smooth open set. We prove that the singular set of any extremal solution of the system \begin{equation*} -\Delta u=\mu e^v , \quad - \Delta v=\lambda The Gel’fand Problem for the Biharmonic Operator • Mathematics • 2013 We study stable and finite Morse index solutions of the equation$${\Delta^2 u = {e}^{u}}$$. If the equation is posed in$${\mathbb{R}^N}$$, we classify radial stable solutions. We then construct A bifurcation diagram of solutions to an elliptic equation with exponential nonlinearity in higher dimensions • Mathematics Proceedings of the Royal Society of Edinburgh: Section A Mathematics • 2017 We consider the following semilinear elliptic equation: where B 1 is the unit ball in ℝ d , d ≥ 3, λ > 0 and p > 0. Firstly, following Merle and Peletier, we show that there exists an eigenvalue λ The Dirichlet problem for$-\Delta \varphi= \mathrm{e}^{-\varphi}$in an infinite sector. Application to plasma equilibria • Mathematics • 2014 We consider here a nonlinear elliptic equation in an unbounded sectorial domain of the plane. We prove the existence of a minimal solution to this equation and study its properties. We infer from Infinitely many nonradial singular solutions of$\Delta u+e^u=0$in$\mathbb{R}^N\backslash\{0\}$,$4\le N\le 10\$
We construct countably infinitely many nonradial singular solutions of the problem $\Delta u+e^u=0\ \ \textrm{in}\ \ \mathbb{R}^N\backslash\{0\},\ \ 4\le N\le 10$ of the form \[